P, Q, and R each try to execute a job and create a report on it.

Question

P, Q, and R each try to execute a job and create a report on it. The individual probabilities for their completion of the jobs are \(\frac{1}{3}\), \(\frac{2}{3}\), and \(\frac{3}{5}\) and the probability for any of them not to finish the report is \(\frac{2}{5}\).

If one can write the report only after finishing the job, then what is the probability that only P and Q will complete their jobs and finish their reports?

The probability that only P and Q will complete their jobs and finish their reports = 1/3*3/5*2/3*3/5*2/5= 1/5*2/5*2/5 = 4/125

IMO B

A.  \frac{4}{625}

B.  \frac{4}{125}

C.  \frac{32}{625}

D.  \frac{4}{25}

E.  \frac{32}{125}

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Nikhil · Accepted Answer

P, Q, and R each try to execute a job and create a report on it.

Completing the job and finishing the report by P, Q and R are independent of each other.

GIven probability for any of them not finishing the report is  \frac{2}{5}

probability for any of them finishing the report is  1-\frac{2}{5} = \frac{3}{5}

To find out the probability that only P and Q will complete their jobs and finish their reports

We need to find

Required probability is the product of three of the following probabilitites

P (P will complete his job  AND  finish the report)
 P (Q will complete his job  AND  finish the report)
 P (R will not complete his job)  OR   P (R will complete his job  AND  not finish the report)

probability that P will complete his job and finish the report =  \frac{1}{3} * \frac{3}{5} = \frac{1}{5}

probability that Q will complete his job and finish the report = \frac{2}{3} * \frac{3}{5} = \frac{2}{5}

probability that R will not complete his job =  1 - \frac{3}{5} = \frac{2}{5}

probability that R will complete his job and not finish the report =  \frac{3}{5} * \frac{2}{5} = \frac{6}{25}

Substituting the above values, we get  (\frac{1}{5})(\frac{2}{5})(\frac{2}{5} + \frac{6}{25}) = \frac{32}{625}

Hence option C

EgmatQuantExpert · Answer

Hey everyone, We will post the solution very soon. Till then, try it one more time and post your analysis.

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loneRanger1050 · Answer

Given that:

Probability that  P  execute the job =  \frac{1}{3} 
Probability that  Q  execute the job =  \frac{2}{3} 
Probability that  R  execute the job =  \frac{3}{5}

Probability that anyone not finishing the report =  \frac{2}{5} 
=> Probability that anyone finish the report = 1 -  \frac{2}{5}  =  \frac{3}{5}

Given that  only  P  and  Q   will execute the job and finish the report. 
So, R  will  not execute the job

Therefore Probability of  R  not executing the job = 1-  \frac{3}{5}  =  \frac{2}{5}

Required probability = (P execute the job and finish report) x ( Q execute the job and finish the report) x (R not executing the job)
 =  (\frac{1}{3} * \frac{3}{5}) ( \frac{2}{3} * \frac{3}{5}) (\frac{2}{5}) 
 =  \frac{1}{5} * \frac{2}{5} * \frac{2}{5} 
 =  \frac{4}{125} 
 Answer = B

Adityasekar · Answer

I did go with C but the OA says B. So here is why it could be B.

P(E) for P and Q to complete their jobs and reports =
     + 
   .

We should remember that when C does not complete this job it also implies that he cannot complete his report.

Substituting the values we get -> (1/3*2/3*3/5)* (3/5*3/5*2/5) = 12/625 ] +  (1/3*2/3*2/5)* (3/5*3/5*2/5) = 8/625 ] =  20/625  =  4/125

Do let me know if my understanding is wrong.

Aditya.
Do give kudos if you find this answer helpful.

Gemelo90 · Answer

The reason why I calculated B and not C is the last part of the question: then what is the probability that only P  and Q will complete their jobs and finish their report  For me completing the job and finishing their report implies that R was not able to complete the job. Therefore, it is not necessary to take into account, that R could have completed the job but not the report.

EgmatQuantExpert · Answer

Solution

Given:  
 • P, Q, and R each try to execute a job and create a report on it
• The individual probabilities for their completion of the jobs are  \frac{1}{3} ,  \frac{2}{3} , and  \frac{3}{5} 
• The probability for any of them not to finish the report is  \frac{2}{5} 
 o Hence, the probability of finishing the report =  (1 – \frac{2}{5}) = \frac{3}{5}  
• One can write the report only after finishing the job

To find:  
 • The probability that only P and Q will complete their jobs and finish their reports

Approach and Working:

it is given that, as per the favourable event, only P and Q will complete their jobs and finish their reports

• The probability that P will execute the job and finish the report =  \frac{1}{3} * \frac{3}{5} = \frac{1}{5} 
• The probability that Q will execute the job and finish the report =  \frac{2}{3} * \frac{3}{5} = \frac{2}{5}

Now, as only P and Q will complete their jobs and finish their reports, it also means, there are two possibilities exist for R

• Either R will not finish the job, and definitely not finish the report (as one cannot finish the report without executing the job)
 o Probability of this event =  (1 – \frac{3}{5}) * \frac{2}{5} = \frac{2}{5} * \frac{2}{5} = \frac{4}{25}  
• Or else, R will execute the job but will not finish the report
 o Probability of this event =  \frac{3}{5} * \frac{2}{5} = \frac{6}{25}

So, we can say,
 • Probability (only P and Q will complete their jobs and finish their reports) = 
P (P completes job & finish report) AND P (Q completes job & finish report) AND  \frac{1}{5} * \frac{2}{5} *   
=  \frac{1}{5} * \frac{2}{5} * \frac{2}{5} 
=  \frac{4}{125}

Hence, the correct answer is option B.

Answer: B

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BARUAH · Answer

Hi expert,

I can not understand why we need to consider the probability of R as the question asks to find the probability that only P and Q will complete their jobs and finish their reports.

Plz help.

Anshup · Answer

Hello expert, request you explain this how do we consider the probability

Sent from my CPH1727 using  GMAT Club Forum mobile app

EgmatQuantExpert · Answer

Hey BARUAH , If you read the question carefully, the favourable event is defined as " only P and Q will complete their jobs and finish their reports." It means we have to ensure that R cannot complete the job and finish the report . For this reason we have to consider the probability of R. Hope this answers your query.

EgmatQuantExpert · Answer

Hey  Anshup ,

Can you provide a little bit more details about you query? The answer we provided takes all the probability values given in the question only.

Gayluk · Answer

Hello  EgmatQuantExpert , thank you for the detailed explanation.

My mistake was to miss the probability of not finishing the report when R did not even finish his job. I thought that since we are told that one cannot even start his report if he didn't finish his job, hence there is no need to multiply 2/5 (1-3/5 - probability that R will not finish his job) with 2/5 (probability of not finishing a report). Please help me understand why we need to take into consideration the probability of not finishing a report when we know that R did jot finish his job, hence he cannot even start his report? Thanks a lot!

Deadpool3 · Answer

Hello  Bunuel

Could you please look at this question.
It seems that answer should be C.

foryearss · Answer

to be honest , I don't understand why we have to multibly (r not finishing the job ) * (r not finishing the report ) in the first case ,since if he didn't finish the job he can't even write the report , so the event will stop there

by multiblying we make the probability smaller than its actual value

serenayong · Answer

Agree. Can any expert shed more light on this?

serenayong · Answer

EgmatQuantExpert 
and in addition, why i think the ans B is impossible. If u draw a probability tree for R, (3/5)(2/5)+(3/5)(3/5)+(2/5)(2/5) does not equal to 1. Since u said its impossible for R to do report if job is not finished, so (3/5)(2/5)+(3/5)(3/5)+(2/5)(2/5)+ (2/5)(3/5) =1 is not possible. so the probability tree should be: (3/5)(2/5)+(3/5)(3/5)+(2/5) =1.

can someone please clarify if my understanding is correct?

ShaikhMoice · Answer

Could you kindly explain why did you multiply 2/5 as if it is sure he will not complete the report there is no need to multiply 2/5.
Thanks in advance.

drbnan · Answer

After sitting with this thread for a few minutes, I am unfortunately convinced that OP is wrong.

The correct answer must in fact be C, 32/625.

If R does not complete the job, he will DEFINITELY not complete the report. Therefore, the probability that he either fails to complete job OR report is 10/25 + 6/25 = 16/25 (absolute probability of failure to complete job) + (conditional probability of completing job but failing report) = 16/25. Therefore, 16/25 is the independent probability of R failing to complete both job and report... a.k.a. NAND (job + report).

You must multiply R's independent combined probability of failure with P and Q's independent conditional probabilities of complete success. (1/3 * 3/5) & (2/3 * 3/5)

This yields (1/5) * (2/5) * (16/25)

*** Where OP goes wrong is that he considers the conditional probability of failing to complete the report after failing to complete the job, rather than accounting for the fact that if the job is not completed, the report cannot be completed, per his instructions.

RustedSteel · Answer

Hey  Nikhil , Actually you went slightly wrong here. There are two events happening simultaneously. 
1. P,Q,R complete/not complete a task.
2. P,Q,R write a report.

and task 2 depend on task 1's completion.
Thus,
Case 1:- We'll calculate prob of P,Q complete the task and R doesn't + Prob of all P,Q,R completing the task. (Since ques asked the prob when only P and Q completed both tasks and report)
P(Event 1) = (P(P success)xP(Q success)xP(R failure)) + ( P(P success)xP(Q success)xP(R success))
 = (1/3 x 2/3 x 3/5) + (1/3 x 2/3 x 2/5) = 10/45 = 2/9
Case 2:- Prob of P, Q completing the report and R doesn't. Here, as we know report can't be written unless task is complete. Hence we won't consider the case where All 3 are writing report. 
Given - prob of not writing report for all three is 2/5. so prob of writing will be 3/5
P(Event 2) = P(P success)xP(Q success)xP(R failure)
 = 3/5 x 3/5 x 2/5 = 18/125

Required prob = P(Event 1) x P(Event 2) = 2/9 x 18/125 = 4/125. Option B

Kinshook · Answer

P, Q, and R each try to execute a job and create a report on it. The individual probabilities for their completion of the jobs are  \frac{1}{3} ,  \frac{2}{3} , and  \frac{3}{5}  and the probability for any of them not to finish the report is  \frac{2}{5} .

If one can write the report only after finishing the job, then what is the probability that only P and Q will complete their jobs and finish their reports?

The probability that only P and Q will complete their jobs and finish their reports = 1/3*3/5*2/3*3/5*2/5= 1/5*2/5*2/5 = 4/125

IMO B

P, Q, and R each try to execute a job and create a report on it.

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GMAT Problem Solving (PS) Questions

P, Q, and R each try to execute a job and create a report on it.

Prep Toolkit

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